I am not a serious mathematician, and I am barely familiar with some of the topics of higher maths. From YouTube videos, I learned about the Graham's Number and Knuth's arrow notation, which is a generalization of the pattern: addition → multiplication → exponentiation ($\uparrow$) → tetration ($\uparrow\uparrow$) → pentation ($\uparrow\uparrow\uparrow$) → etc.
Independently, I've realized that:
$$2\uparrow\uparrow\uparrow2 = 2\uparrow\uparrow2 = 2\uparrow2 = 2\times2 = 2+2 = 4$$
… and it looks like the identity holds for higher orders of operation (above pentation); in fact, it is not hard to see that it holds for all orders, with any number of arrows.
Is it a well-known fact? Is it a particularly interesting one? Can $2$ be an identity of some kind, just like $0$ is an additive identity, and $1$ is a multiplicative identity?